We present a new algorithm for computing ae where a in
GF(2k) and e is a positive integer. The proposed algorithm is
more suitable for implementation in software, and relies on the Montgomery
multiplication in GF(2k). The speed of the exponentiation
algorithm largely depends on the availability of a fast method for multiplying
two polynomials of length w defined over GF(2). The theoretical analysis and our
experiments indicate that the proposed exponentiation method is at least 6 times
faster than the exponentiation method using the standard multiplication when w=8.
Furthermore, the availability of a 32-bit GF(2) polynomial multiplication
instruction on the underlying processor would make the new exponentiation
algorithm up to 37 times faster.